Question: Convert $427_8$ to base 5.
Solution: First we convert to base 10, obtaining $427_8 = 4 \cdot 8^2 + 2 \cdot 8^1 + 7 \cdot 8^0 = 279.$ Then we convert 279 to base 5 to get  \begin{align*}
279 &= 2 \cdot 5^3 + 29 \\
&= 2 \cdot 5^3 + 1 \cdot 5^2 + 4 \\
&= 2 \cdot 5^3 + 1 \cdot 5^2 + 4 \cdot 5^0 \\
&=\boxed{2104_5}.
\end{align*}